Archive for the ‘Analysis’ Category
Inequalities.
tuesday, february 9th, 2010This post shows a diagram, listing a lot of inequalities and showing implications between them.
The Hasse derivative, part II: Multivariate partial Hasse derivatives.
friday, october 2nd, 2009Following a suggestion by A. Maevskiy, we show how the Hasse derivative can be extended to partial Hasse derivative in arbitrary multivariate polynomial rings. We show multivariate versions of Taylor’s Formula, of the Identity Theorem, and of the Generalized Leibnitz Rule.
Functional Calculus in Linear Algebra, the Jordan Decomposition Reloaded and Cayley-Hamilton’s Theorem.
thursday, august 13th, 2009We explain the aims of functional calculus and specialize to polynomials evaluated at endomorphisms. We reconsider the Jordan decomposition and prove it with more generality. Then, we discuss Taylor expansion in the nilpotent part for endomorphisms with separable minimal polynomials, and prove Cayley-Hamilton again for arbitrary fields.
The Hasse derivative.
wednesday, august 12th, 2009In real and complex analysis, the Taylor series expansion is a very important tool. For polynomials over arbitrary unitary rings, it is possible to define a derivative which behaves similar to the usual derivative; unfortunately, the Identity Theorem and Taylor’s formula do not transfer to this new situation. Fortunately, there exists a different definition of derivatives for these cases, namely the Hasse derivative. Not only does it gives a Identity Theorem and Taylor’s formula back, but also allows to write other identities in a simpler way.
A Topological Proof of the Cayley-Hamilton Theorem over all Commutative Unitary Rings.
monday, may 4th, 2009We want to give a proof of the Cayley-Hamilton Theorem for all commutative rings with unity, which first reduces to the case of the field of complex numbers and then applies a topological argument.
Fundamental Theorem of Algebra.
monday, may 4th, 2009We want to give a proof of the Fundamental Theorem of Algebra using methods from Complex Analysis, in particular Liouville’s Theorem.
